The Volume of the Universe after Inflation and de Sitter Entropy

TL;DR

The paper provides a detailed stochastic-inflation analysis to derive the full probability distribution of the reheating-volume ρ(V) for both non-eternal and eternal inflation, using a discrete-to-continuum mapping to a nonlinear differential equation for the Laplace transform f(φ;s0). It shows that the transition at Ω=1 marks the onset of eternal inflation, and that the quantum bound on finite-volume production is sharp, with violations being super-exponentially suppressed as e^{−e^{S_{dS}}}, implying a coefficient c≈1/2 in the bound. Realistic considerations, including barriers and slow-roll corrections, modify the tails but preserve the overall structure: a peaked bulk around the classical trajectory, with a tail controlled by Ω and barrier position. The results provide both a consistency check for de Sitter complementarity and a concrete, calculable framework for understanding the geometry and measure problems in eternal inflation.

Abstract

We calculate the probability distribution for the volume of the Universe after slow-roll inflation both in the eternal and in the non-eternal regime. Far from the eternal regime the probability distribution for the number of e-foldings, defined as one third of the logarithm of the volume, is sharply peaked around the number of e-foldings of the classical inflaton trajectory. At the transition to the eternal regime this probability is still peaked (with the width of order one e-folding) around the average, which gets twice larger at the transition point. As one enters the eternal regime the probability for the volume to be finite rapidly becomes exponentially small. In addition to developing techniques to study eternal inflation, our results allow us to establish the quantum generalization of a recently proposed bound on the number of e-foldings in the non-eternal regime: the probability for slow-roll inflation to produce a finite volume larger than e^(S_dS/2), where S_dS is the de Sitter entropy at the end of the inflationary stage, is smaller than the uncertainty due to non-perturbative quantum gravity effects. The existence of such a bound provides a consistency check for the idea of de Sitter complementarity.

Figures (11)

• Figure 1: Set of space-like slices covering both the exterior and the interior of: a) a black hole (left) and b) an eternal inflating universe (right).
• Figure 2: Typical shape for the probability distribution of the volume $\rho(V)$. For small volumes the behavior is gaussian with the number of $e$-foldings ($\rho\sim e^{-c(N-\overline N)^2}$); for volumes larger than the average value $\overline V$, $\rho(V)$ follows a power law in the volume ($\rho\sim 1/V^\alpha$) that eventually turns into an exponential law ($\rho \sim e^{-{\rm const}\cdot V}$) at large enough volumes ($V\gtrsim V_b$). When $\Omega<1$ the exponential tail starts earlier at $V\simeq V_\epsilon=e^{\pi/(2\sqrt{1-\Omega})}$.
• Figure 3: The branching process.
• Figure 4: Left: Plot of $F_1(s)$ restricted to the hypercube $I_L$ and with $s_0=1$, for large $p$ (thick curve). The only fixed point in the unit cube is $s=1$. Further applications of $F_1$ (thinner curves) drive the curve to the $F_{\infty} = 1$ line. Right: For smaller $p$'s a new fixed point $s_f$ enters the unit cube. Now the limiting line is $F_{\infty} = s_f$.
• Figure 5: The 2-site branching process.